Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 102

, , ,

log S = log S + log(2ʌ) + log (f) = log S + log(2ʌ) log (T)

a v v

and as a correlation:

, ,

log S = log S log(2ʌ) log (f) = log S log(2ʌ) + log (T)

d v v

Thus, it is quite easy, from a given accelerogram, to derive response spectra.
However, a given response spectrum (for a given damping value) corresponds to an
infinite number of time histories with the same acceleration and displacement
maxima but different durations and phases. The repartition of the energy arrival in
time is not constrained by the response spectrum data. The absence of equivalence
constitutes a great handicap when non-linear time studies have to be led, and using
even adapted accelerograms to a given response spectrum is a difficult task which
can only be performed by seismologists if they are to appear realistic.

Elastic response spectra are often used in paraseismic engineering because, as a
first estimate, simple structures can be compared to a 1-dof oscillator with a known
period and damping. Caution must be urged in this regard, as the conventional
formula T = N/10 (with N being the number of floors) has been imported from the
USA where most buildings are framework constructions and are not applicable in
France, where the prevailing structures have stiffer walls for which the formula T =
N/25 is more appropriate. The motions of the center of gravity can be reasonably
well estimated, as long as the structure is assumed to have a linear elastic behavior.

This last hypothesis is obviously not true when the structure has been seriously
damaged: that is why other non-linear spectra have been developed, among which
the simplest correspond to perfect elasto-plastic behavior. They involve the
introduction of an additional parameter, the ductility demand, (P), which defines the
relationship between the maximum displacement of the elasto-plastic structure and
that of the associated elastic structure (with the same low acceleration stiffness and
damping).

As estimating the dynamic response of non-linear systems is difficult, it has
recently been proposed that dynamic non-linear analysis is replaced by a static non-
linear analysis, called “push-over analysis”. The gross result of this is a curve that
links the applied horizontal force to the displacement. Procedures have also been
proposed for converting elastic response spectra into “stress-strain” curves, or to
transform the latter into “S a - S d ” curves, linking force to acceleration by means of the
modal mass.